OFFSET
0,1
COMMENTS
The function product{1 + x^k, k >= 1} has two inflection points: (-0.78983..., 0.17671...) and (-0.23233..., 0.80084...).
EXAMPLE
x = -0.2323377316586730804879725477175353981424044769...
MATHEMATICA
f[x_] := f[x] = Product[(1 + x^k), {k, 1, 1000}];
p[x_, z_] := Sum[n/(x + x^(1 - n)), {n, z}]^2 + Sum[(n*x^(n - 2)*(n - x^n - 1))/(1 + x^n)^2, {n, z}];
Plot[f[x], {x, -1, 1}] (* plot showing 2 infl. pts. *)
t = x /. FindRoot[p[x, 1000], {x, -0.8}, WorkingPrecision -> 100] (* A257394 *)
u = f[t] (* A257395 *)
v = x /. FindRoot[p[x, 200], {x, -0.3}, WorkingPrecision -> 100] (* A257396 *)
w = f[v] (* A257397 *)
RealDigits[t, 10][[1]] (* A257394 *)
RealDigits[u, 10][[1]] (* A257395 *)
RealDigits[v, 10][[1]] (* A257396 *)
RealDigits[w, 10][[1]] (* A257397 *)
(* Peter J. C. Moses, Apr 21 2015 *)
digits = 104; QP = QPochhammer; QPP[x_] := With[{dx = 10^-digits}, (QP[-1, x+dx] - QP[-1, x-dx])/(4*dx)]; x0 = x /. NMinimize[{QPP[x], -1/2 < x < 0}, x, WorkingPrecision -> 4 digits][[2]]; RealDigits[x0, 10, digits] // First (* Jean-François Alcover, Nov 19 2015 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Apr 22 2015
EXTENSIONS
More digits from Jean-François Alcover, Nov 19 2015
STATUS
approved